Let \(I \subseteq \lbrace 1,\dots ,k\rbrace \)  of size \(|I|<r\) , and \(f \in W_{\kappa r}\) . It is sufficient to show \(f a_I = 0\)  for \(f\)  in every one of the simple \(A\) -modules in \(W_{\kappa r}\)  as given by the direct sum in Definition REF , and the general case follows by linearity. Hence let \(f \in \Theta [T] S^{\lambda }\)  with a partition \(\lambda \vdash k\)  such that \(\lambda _{\text{\bfseries \ttfamily a}}= k-r\) , where \(T\)  is a semistandard table with shape \(\lambda \)  and composition \(\kappa \) , see §REF . Moreover, since the map \(\Theta [T]\)  is equivariant to \(A\) 's action, it is enough to show \(f^{\prime } a_I = 0\)  for \(f^{\prime } \in S^{\lambda }\)  such that \(f = \Theta [T]f^{\prime }\) . Recalling the definition \(S^{\lambda } = \alpha _{\lambda } b_{\lambda } A\) , it is left to prove \(b_{\lambda } \sigma a_I = 0\)  for every \(\sigma \in S_k\) . Note that \(\sigma a_I = a_{(\sigma I)} \sigma \)  where \(\sigma I = \lbrace \sigma (i) \mid i \in I\rbrace \)  and \(|\sigma I|<r\)  as well, so the general case would follow from showing \(b_{\lambda }a_I = 0\) .
These elements were defined as \(a_I = \sum _{\tau \in \operatorname{stab}I} \tau \)  and \(b_{\lambda } = \sum _{\tau \in Q_{\lambda }} \mathrm {sign}(\tau ) \tau \) , and the subgroup \(Q_{\lambda }\)  as all the permutations of \(\lbrace 1,\dots ,k\rbrace \)  permuting the numbers within certain \(\lambda _{\text{\bfseries \ttfamily a}}\)  subsets that compose this set. The remaining argument is essentially Lemma 4.23 of [1]}. Since \(|I|<r = k-\lambda _{\text{\bfseries \ttfamily a}}\) , there exists \(i,j \notin I\)  such that the transposition \((ij) \in Q_{\lambda } \cap \operatorname{stab}I\) . Therefore, \(b_{\lambda } (ij) = -b_{\lambda }\)  and \((ij)a_I = a_I\)  so that \(b_{\lambda }a_I = -b_{\lambda }a_I = 0\)  as required.
